"To start the hare" is a hunting term that means to begin a hunt. The hare is the animal being chased, and the hunt begins when the hare is released from its starting point.
The meaning of the idiom "to start the hare" is to initiate something, to begin a process or activity.
The phrase "to start a hare" is incorrect and does not have a specific meaning. It's likely a mishearing or misunderstanding of the original idiom.